Notes, summaries, assignments, exams, and problems for Mathematics

Stock Valuation and WACC Examples

Problem 1: Constant Dividend Growth

Thomas Brothers is expected to pay a $0.35 per share dividend at the end of the year (D1 = $0.35). The dividend is expected to grow at a constant rate of 6% a year. The required rate of return is 13%. What is the stock’s current price?

P0 = D1 / (r – g) = 0.35 / (0.13 - 0.06) = $5.00

Problem 2: Current Stock Price with Constant Growth

A stock just paid a dividend of D0 = $1.50. The required rate of return is r = 10%, and the constant growth rate is g = 4.0%. What is the current stock price?

P0 = D0(1+g) / (r – g) = 1.50(1 + 0.04) / (0.10 - 0.04) = $26.00

Problem 3: Expected Stock Price in One Year

Using your answer for problem 2, what is the expected stock price in one year... Continue reading "Calculating Stock Prices and Weighted Average Cost of Capital (WACC)" »

Actuarial Science Practice Problems

Problem 11: Investment Option Comparison

11. $50,000 is invested in two options:

• Option A: Invested into a fund with an annual effective rate i.
• Option B: Purchase of an annuity-immediate with 24 level annual payments at a rate of 10%.

Option B is equivalent to Option A with an interest rate of 5%. The accumulated value for Option A is 50,000(1+i)²⁴. For Option B: * = Option A. Find i.

Problem 14: Monthly Deposits and Future Value

14. Smith deposits $1,000 at the end of each month with a nominal rate i⁽¹²⁾ = 0.12. The first deposit is on 1/31/2010, and the last is on 12/31/2034. From 2035 to 2059, the accumulated value Y is:

Problem 15: Interest Rate Changes and Balances

15. Since 6/30/2014, a deposit of $100... Continue reading "Actuarial Mathematics: Annuity and Loan Practice Problems" »

Linguistic Exercise: Defining Corners

3.4.3. Match the definitions to the words.

Definitions:

• 1) Convex area where two edges, sides, or surfaces of something join.
• 2) Concave area where two edges, sides, or surfaces of something join.
• 3) Area where two edges, sides, or surfaces of something join.

Words:

• a) Corner (English)
• b) Esquina (Spanish)
• c) Rincón (Spanish)

Exercise Solutions:

1. Convex area: Esquina
2. Concave area: Rincón
3. General area: Corner

Figure Matching:

• Fig A: Corner / Esquina
• Fig B: Corner / Rincón

Semantic Analysis: Distinctive Features

Discuss the following issue: Is "concave" a distinctive semantic feature (seme) in English? And in Spanish?

According to structuralism, distinctive features are those which help us differentiate the units within the... Continue reading "Semantic Analysis of Corner in English and Spanish" »

Core Probability Definitions and Formulas

• Independence: Events A and B are independent if P(A ∩ B) = P(A)P(B).
• Mutual Exclusivity: Events A and B are mutually exclusive if P(A ∩ B) = 0.

Key Relationships

• If P(A) + P(B) > 1, then A and B are not mutually exclusive. (True)
• If P(A) + P(B) = 1, A and B are not necessarily mutually exclusive. (False, unless A and B are complements)

Conditional Probability and Addition Rule

• Multiplication Rule: P(A ∩ B) = P(A) P(B|A).
• Conditional Probability Definition: P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0.
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

Advanced Probability Relationships

Bayes' Theorem Setup

Given P(A|B) = p, P(B|A) = q, and P(A) = r. Using the definition of conditional probability:... Continue reading "Conditional Probability and Discrete Distributions" »

Limits of Sequences

We will often need to specify that a number x is close to another number a. However, this doesn’t mean anything unless we specify how close. If ε is a positive number, then the statement “x is within ε of a” does have meaning. It means that the distance between x and a is less than ε – that is, |x − a| ε.

Theorem 2.1.1

If x, y, a, and ε are real numbers with ε > 0, then (a) |y|.

Theorem 2.1.2 (Triangle Inequality)

If a and b are real numbers, then (a) |a + b| ≤ |a| + |b|; and (b) ||a| − |b| ≤ |a − b|. Proof: For part (a), we observe that −|a| ≤ a ≤ |a| and −|b| ≤ b ≤ |b|. If we add these inequalities, the result is
−(|a| + |b|) ≤ a + b ≤ |a| + |b|. By the preceding theorem (with “&... Continue reading "Understanding Limits of Sequences in Mathematics" »

Geometry

Angles

Adjacent angles - that have a common vertex and side, but no common interior points

Angle - formed by two rays with a common endpoint called the vertex

Complementary angles - whose measures add up to 90°

Congruent angles - that have the same measure

Supplementary angles - two angles whose measures add up to 180°

Vertical angles - the nonadjacent angles formed by two intersecting lines

Lines

Midpoint - the point that divides a segment into two congruent segments

Parallel lines - in a plane that never meet

Perpendicular lines - that intersect at 90° angles

Transversal - a line that intersects any two or more other lines

Triangles

Triangle Inequality Theorem - the sum of the lengths of any two sides of a triangle is greater than the length... Continue reading "Essential Geometry and Algebra Glossary" »

Statistics: Key Concepts and Methods

1. Statistics is a collection of methods for planning experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions.

2. Statistical methods consist of a series of procedures for managing, analyzing, and collecting qualitative and quantitative research data.

3. A population includes all objects of interest, whereas a sample is only a portion of the population.

Example: The population could be all students of a postgraduate program, and a representative sample of this population could be 400 students from different studies of the program.

Understanding Variables in Statistics

4. A variable is any characteristic, number, or quantity that can be measured... Continue reading "Key Concepts and Methods in Statistical Analysis" »

Unit 6: Costing and Production Systems

1. Production Systems

The two common cost accumulation systems are job order costing and process costing. The election to use one or the other of these systems depends primarily on the nature of the product and the manufacturing process by which it is created. Process production and job-order production are the extreme possibilities when considering the heterogeneity of products. The difference between order accounting and section accounting is the aim of the cost system, and the difference between process costing and job costing is the method to assign the cost to products.

1.1. Job Order Costing System

Job order costing suits the processes where it is necessary to assign costs to a specific amount of production... Continue reading "Costing and Production Systems: Job Order and Process Costing" »

Total Cost = w₁x₁ + w₂x₂ + … + w_nx_n

Isocost function:

x₂ = -w₁x₁/w₂ + tc/w₂ where the slope is: -w₁/w₂ ratio of relative factor prices with negative sign

Cost minimization mate: -w₁/w₂ = -MP₁/MP₂ (isoquant curve)

Cost minimization implies producing a certain amount of output (y > 0) with the lowest possible cost. As with the maximization of profits, the firm is getting the most out of the resources it is using.

What is the difference between cost minimization and profit maximization? Cost minimization is a necessary condition for the maximization of profits, but not the other way around. For example, if you bring costs down to zero then output and profits will be zero too, failing to maximize profits while minimizing costs. As before, when... Continue reading "Cost Minimization and Profit Maximization in Economics" »

Pocket money isn’t a controversial topic nowadays.

If there have been a lot of debates over pocket money.

Pocket money used to be a controversial topic.

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Pocket money has always been a controversial topic.

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The debate regarding pocket money nowadays is about how much money should be given.

If today’s debate rages about something entirely different.

The debate regarding pocket money nowadays isn’t about how much money should be given.

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The debate regarding pocket money nowadays is about at what age money should be given.

If

Twelve-year-old will be able to use the new debit card.

A new debit card has just been launched for teenagers as young as 13.

Thirteen-year-olds won’t be able to use the credit card.

Teenagers over 13 will be able to use the... Continue reading "The Future of Pocket Money: Debit Cards and Financial Management" »